The topological crystals shown in Fig. REF  (a), (b) are real space limits of weak topological phases corresponding to stacks of twofold rotation symmetric one dimensional topological phases.
Figure REF  (f) shows that the topological crystal shown in Fig. REF  (a) can be trivialized by a doubling of the unit cell in \(x\)  direction. However, it is invariant under a doubling of the unit cell in \(y\)  direction as shown in Figure REF  (e). A similar argument holds for the topological crystal shown in Fig. REF  (b). The topological crystal shown in Fig. REF  (d) is the real space limit of a strong second-order topological superconductor. It is invariant under a redefinition of the unit cell in both \(x\)  and \(y\)  direction, as seen in Fig. REF  (g) and (h), respectively. Due to the equivalence relation, the strong second-order topological phase has \(\mathbb {Z}_2\)  character, i.e. the topological crystal with topological invariants \((0,2,-2)\)  can be adiabatically deformed to the trivial crystal \((0,0,0)\)  up to an atomic limit or decorated \(d-2\)  cells parallel to the rotation axis. In case decorated \(d-2\)  cells remain, the resulting topological crystal may be a strong third order topological phase, as expected from the \(K\) -theoretic results from Ref. [1]}.
